Problem: $\dfrac{ -5n + 7p }{ 2 } = \dfrac{ 7n + 6q }{ 6 }$ Solve for $n$.
Explanation: Multiply both sides by the left denominator. $\dfrac{ -5n + 7p }{ {2} } = \dfrac{ 7n + 6q }{ 6 }$ ${2} \cdot \dfrac{ -5n + 7p }{ {2} } = {2} \cdot \dfrac{ 7n + 6q }{ 6 }$ $-5n + 7p = {2} \cdot \dfrac { 7n + 6q }{ 6 }$ Multiply both sides by the right denominator. $-5n + 7p = 2 \cdot \dfrac{ 7n + 6q }{ {6} }$ ${6} \cdot \left( -5n + 7p \right) = {6} \cdot 2 \cdot \dfrac{ 7n + 6q }{ {6} }$ ${6} \cdot \left( -5n + 7p \right) = 2 \cdot \left( 7n + 6q \right)$ Distribute both sides ${6} \cdot \left( -5n + 7p \right) = {2} \cdot \left( 7n + 6q \right)$ $-{30}n + {42}p = {14}n + {12}q$ Combine $n$ terms on the left. $-{30n} + 42p = {14n} + 12q$ $-{44n} + 42p = 12q$ Move the $p$ term to the right. $-44n + {42p} = 12q$ $-44n = 12q - {42p}$ Isolate $n$ by dividing both sides by its coefficient. $-{44}n = 12q - 42p$ $n = \dfrac{ 12q - 42p }{ -{44} }$ All of these terms are divisible by $2$ Divide by the common factor and swap signs so the denominator isn't negative. $n = \dfrac{ -{6}q + {21}p }{ {22} }$